3.1.91 \(\int \frac {9-21 x+6 x^2+(3-6 x) \log (4)+(-9+9 x+(-3+3 x) \log (4)) \log (-x+x^2)+(-9+15 x-7 x^2+x^3+(-6+8 x-2 x^2) \log (4)+(-1+x) \log ^2(4)) \log ^2(-x+x^2)}{(-9+15 x-7 x^2+x^3+(-6+8 x-2 x^2) \log (4)+(-1+x) \log ^2(4)) \log ^2(-x+x^2)} \, dx\)

Optimal. Leaf size=25 \[ -1+x+\frac {3 x}{(3-x+\log (4)) \log \left (-x+x^2\right )} \]

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Rubi [F]  time = 1.17, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {9-21 x+6 x^2+(3-6 x) \log (4)+(-9+9 x+(-3+3 x) \log (4)) \log \left (-x+x^2\right )+\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )}{\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 - 21*x + 6*x^2 + (3 - 6*x)*Log[4] + (-9 + 9*x + (-3 + 3*x)*Log[4])*Log[-x + x^2] + (-9 + 15*x - 7*x^2 +
x^3 + (-6 + 8*x - 2*x^2)*Log[4] + (-1 + x)*Log[4]^2)*Log[-x + x^2]^2)/((-9 + 15*x - 7*x^2 + x^3 + (-6 + 8*x -
2*x^2)*Log[4] + (-1 + x)*Log[4]^2)*Log[-x + x^2]^2),x]

[Out]

x + Defer[Int][(-3 + 6*x)/((1 - x)*(3 - x + Log[4])*Log[(-1 + x)*x]^2), x] + (9 + Log[64])*Defer[Int][1/((3 -
x + Log[4])^2*Log[(-1 + x)*x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6 x^2-9 \left (1+\frac {2 \log (2)}{3}\right )+3 x (7+\log (16))-(-3 (3+\log (4))+x (9+\log (64))) \log ((-1+x) x)-(-1+x) (3-x+\log (4))^2 \log ^2((-1+x) x)}{(1-x) (3-x+\log (4))^2 \log ^2((-1+x) x)} \, dx\\ &=\int \left (1+\frac {-9-6 x^2+3 x (7+\log (16))-\log (64)}{(1-x) (3-x+\log (4))^2 \log ^2((-1+x) x)}+\frac {3 (3+\log (4))-x (9+\log (64))}{(1-x) (3-x+\log (4))^2 \log ((-1+x) x)}\right ) \, dx\\ &=x+\int \frac {-9-6 x^2+3 x (7+\log (16))-\log (64)}{(1-x) (3-x+\log (4))^2 \log ^2((-1+x) x)} \, dx+\int \frac {3 (3+\log (4))-x (9+\log (64))}{(1-x) (3-x+\log (4))^2 \log ((-1+x) x)} \, dx\\ &=x+(9+\log (64)) \int \frac {1}{(3-x+\log (4))^2 \log ((-1+x) x)} \, dx+\int \frac {-3+6 x}{(1-x) (3-x+\log (4)) \log ^2((-1+x) x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 46, normalized size = 1.84 \begin {gather*} x-\frac {x \left (9-21 x+6 x^2-3 x \log (16)+\log (64)\right )}{(-1+2 x) (-3+x-\log (4))^2 \log ((-1+x) x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 - 21*x + 6*x^2 + (3 - 6*x)*Log[4] + (-9 + 9*x + (-3 + 3*x)*Log[4])*Log[-x + x^2] + (-9 + 15*x - 7
*x^2 + x^3 + (-6 + 8*x - 2*x^2)*Log[4] + (-1 + x)*Log[4]^2)*Log[-x + x^2]^2)/((-9 + 15*x - 7*x^2 + x^3 + (-6 +
8*x - 2*x^2)*Log[4] + (-1 + x)*Log[4]^2)*Log[-x + x^2]^2),x]

[Out]

x - (x*(9 - 21*x + 6*x^2 - 3*x*Log[16] + Log[64]))/((-1 + 2*x)*(-3 + x - Log[4])^2*Log[(-1 + x)*x])

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fricas [A]  time = 0.57, size = 45, normalized size = 1.80 \begin {gather*} \frac {{\left (x^{2} - 2 \, x \log \relax (2) - 3 \, x\right )} \log \left (x^{2} - x\right ) - 3 \, x}{{\left (x - 2 \, \log \relax (2) - 3\right )} \log \left (x^{2} - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-1)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)*log(x^2-x)^2+(2*(3*x-3)*log(2)+9*x-9)*l
og(x^2-x)+2*(-6*x+3)*log(2)+6*x^2-21*x+9)/(4*(x-1)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)/log(x^2-
x)^2,x, algorithm="fricas")

[Out]

((x^2 - 2*x*log(2) - 3*x)*log(x^2 - x) - 3*x)/((x - 2*log(2) - 3)*log(x^2 - x))

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giac [A]  time = 0.83, size = 40, normalized size = 1.60 \begin {gather*} x - \frac {3 \, x}{x \log \left (x^{2} - x\right ) - 2 \, \log \relax (2) \log \left (x^{2} - x\right ) - 3 \, \log \left (x^{2} - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-1)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)*log(x^2-x)^2+(2*(3*x-3)*log(2)+9*x-9)*l
og(x^2-x)+2*(-6*x+3)*log(2)+6*x^2-21*x+9)/(4*(x-1)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)/log(x^2-
x)^2,x, algorithm="giac")

[Out]

x - 3*x/(x*log(x^2 - x) - 2*log(2)*log(x^2 - x) - 3*log(x^2 - x))

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maple [A]  time = 0.18, size = 27, normalized size = 1.08

method	result	size
risch	\(x +\frac {3 x}{\ln \left (x^{2}-x \right ) \left (2 \ln \relax (2)+3-x \right )}\)	\(27\)
norman	\(\frac {\left (4 \ln \relax (2)^{2}+12 \ln \relax (2)+9\right ) \ln \left (x^{2}-x \right )+3 x -\ln \left (x^{2}-x \right ) x^{2}}{\left (2 \ln \relax (2)+3-x \right ) \ln \left (x^{2}-x \right )}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*(x-1)*ln(2)^2+2*(-2*x^2+8*x-6)*ln(2)+x^3-7*x^2+15*x-9)*ln(x^2-x)^2+(2*(3*x-3)*ln(2)+9*x-9)*ln(x^2-x)+2
*(-6*x+3)*ln(2)+6*x^2-21*x+9)/(4*(x-1)*ln(2)^2+2*(-2*x^2+8*x-6)*ln(2)+x^3-7*x^2+15*x-9)/ln(x^2-x)^2,x,method=_
RETURNVERBOSE)

[Out]

x+3*x/ln(x^2-x)/(2*ln(2)+3-x)

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maxima [B]  time = 0.60, size = 64, normalized size = 2.56 \begin {gather*} \frac {{\left (x^{2} - x {\left (2 \, \log \relax (2) + 3\right )}\right )} \log \left (x - 1\right ) + {\left (x^{2} - x {\left (2 \, \log \relax (2) + 3\right )}\right )} \log \relax (x) - 3 \, x}{{\left (x - 2 \, \log \relax (2) - 3\right )} \log \left (x - 1\right ) + {\left (x - 2 \, \log \relax (2) - 3\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-1)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)*log(x^2-x)^2+(2*(3*x-3)*log(2)+9*x-9)*l
og(x^2-x)+2*(-6*x+3)*log(2)+6*x^2-21*x+9)/(4*(x-1)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)/log(x^2-
x)^2,x, algorithm="maxima")

[Out]

((x^2 - x*(2*log(2) + 3))*log(x - 1) + (x^2 - x*(2*log(2) + 3))*log(x) - 3*x)/((x - 2*log(2) - 3)*log(x - 1) +
(x - 2*log(2) - 3)*log(x))

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mupad [B]  time = 1.48, size = 712, normalized size = 28.48 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2 - x)*(9*x + 2*log(2)*(3*x - 3) - 9) - 2*log(2)*(6*x - 3) - 21*x + log(x^2 - x)^2*(15*x + 4*log(2)
^2*(x - 1) - 2*log(2)*(2*x^2 - 8*x + 6) - 7*x^2 + x^3 - 9) + 6*x^2 + 9)/(log(x^2 - x)^2*(15*x + 4*log(2)^2*(x
- 1) - 2*log(2)*(2*x^2 - 8*x + 6) - 7*x^2 + x^3 - 9)),x)

[Out]

x + ((3*x)/(2*log(2) - x + 3) - (x*log(x^2 - x)*(log(64) + 9)*(x - 1))/((2*x - 1)*(12*log(2) - 6*x - 4*x*log(2
) + 4*log(2)^2 + x^2 + 9)))/log(x^2 - x) - ((1386*log(2) + 1017*log(4) + 72*log(8) - 404*log(512) - 6*log(2)*l
og(4) + 300*log(4)*log(8) - 232*log(4)*log(512) - 444*log(2)*log(4)^2 - 1248*log(2)^2*log(4) - 96*log(2)*log(4
)^3 + 248*log(4)^2*log(8) + 76*log(4)^3*log(8) + 8*log(4)^4*log(8) - 32*log(4)^2*log(512) - 1356*log(2)^2 + 93
6*log(4)^2 + 306*log(4)^3 + 36*log(4)^4 - 408*log(2)^2*log(4)^2 - 48*log(2)^2*log(4)^3)/(2*(150*log(4) + 60*lo
g(4)^2 + 8*log(4)^3 + 125)) - (x*(552*log(2) + 2169*log(4) + 204*log(8) - 228*log(512) - 810*log(2)*log(4) + 4
64*log(4)*log(8) - 72*log(4)*log(512) - 612*log(2)*log(4)^2 - 1008*log(2)^2*log(4) - 96*log(2)*log(4)^3 + 312*
log(4)^2*log(8) + 84*log(4)^3*log(8) + 8*log(4)^4*log(8) - 1092*log(2)^2 + 1296*log(4)^2 + 342*log(4)^3 + 36*l
og(4)^4 - 360*log(2)^2*log(4)^2 - 48*log(2)^2*log(4)^3 + 1125))/(150*log(4) + 60*log(4)^2 + 8*log(4)^3 + 125)
+ (x^2*(432*log(2) + 1404*log(4) + 214*log(8) - 48*log(512) + 72*log(2)*log(4) + 288*log(4)*log(8) + 120*log(4
)^2*log(8) + 16*log(4)^3*log(8) - 72*log(2)^2 + 540*log(4)^2 + 72*log(4)^3 + 1125))/(150*log(4) + 60*log(4)^2
+ 8*log(4)^3 + 125))/(6*log(4) - x*(14*log(4) + 2*log(4)^2 + 24) + x^2*(4*log(4) + 13) + log(4)^2 - 2*x^3 + 9)
- (atan(((4*x - (1300*log(4) + 720*log(4)^2 + 176*log(4)^3 + 16*log(4)^4 + 875)/(150*log(4) + 60*log(4)^2 + 8
*log(4)^3 + 125))*(150*log(4) + 60*log(4)^2 + 8*log(4)^3 + 125)*(36*log(8) - 432*log(2) + 48*log(512) - log(4)
*(72*log(2) - 12*log(8) + 54) + 72*log(2)^2)*1i)/((2*log(4) + 5)^4*(36*log(8) - 54*log(4) - 432*log(2) + 48*lo
g(512) - 72*log(2)*log(4) + 12*log(4)*log(8) + 72*log(2)^2)))*(36*log(8) - 432*log(2) + 48*log(512) - log(4)*(
72*log(2) - 12*log(8) + 54) + 72*log(2)^2)*2i)/(2*log(4) + 5)^4

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sympy [A]  time = 0.20, size = 19, normalized size = 0.76 \begin {gather*} x - \frac {3 x}{\left (x - 3 - 2 \log {\relax (2 )}\right ) \log {\left (x^{2} - x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-1)*ln(2)**2+2*(-2*x**2+8*x-6)*ln(2)+x**3-7*x**2+15*x-9)*ln(x**2-x)**2+(2*(3*x-3)*ln(2)+9*x-9)
*ln(x**2-x)+2*(-6*x+3)*ln(2)+6*x**2-21*x+9)/(4*(x-1)*ln(2)**2+2*(-2*x**2+8*x-6)*ln(2)+x**3-7*x**2+15*x-9)/ln(x
**2-x)**2,x)

[Out]

x - 3*x/((x - 3 - 2*log(2))*log(x**2 - x))

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